We present a three-dimensional constitutive model for NiTi polycrystalline shape memory alloys exhibiting transformations between three solid phases (austenite, R-phase, martensite). The ‘full modelling sequence’ comprised of formulation of modelling assumptions, construction of the model, mathematical analysis and numerical implementation and validation is presented. Namely, by formulating micromechanics-inspired modelling assumptions we concentrate on describing the dissipation mechanism: a refined form of this description makes our model especially useful for complex loading paths. We then embed the model into the so-called energetic framework (extended to our case) while taking advantage of describing the dissipation mechanism through the so-called dissipation distance. We prove the existence of energetic solutions to our model by a backward Euler scheme. This is then implemented into finite element software, and numerical simulations compared with experiments are also presented.
BhattacharyaK. Microstructure of martensite: Why it forms and how it gives rise to the shape-memory effect. Oxford: Oxford University Press, 2003.
2.
BallJMJamesRD. Fine phase mixtures as minimizers of energy. Arch Rat Mech Anal1987; 100(1): 13–52.
3.
PitteriMZanzottoG. Continuum models for phase transitions and twinning in crystals (Applied Mathematics, vol. 19). Boca Raton, FL: CRC Press, 2003.
4.
MachadoLGSaviMA. Medical applications of shape memory alloys. Braz J Med Biol Res2003; 36: 683–691.
5.
WilliamsEElahiniaMH. An automotive SMA mirror actuator: Modeling, design and experimental evaluation. J Intell Mater Syst Struct2008; 19: 1425–1434.
6.
VokounDSedlákPFrostM. Velcro-like fasteners based on NiTi micro-hook arrays. Smart Mater Struct2011; 20: 085027.
7.
LagoudasDCEntchevPBPopovP. Shape memory alloys. Part II: Modeling of polycrystals. Mech Mater2006; 38: 430–462.
8.
PatoorELagoudasDCEntchevPB. Shape memory alloys, Part I: General properties and modeling of single crystals. Mech Mater2006; 38: 391–429.
9.
KhandelwalABuravallaV. Models for shape memory alloy behavior: An overview of modeling approaches. Int J Struct Changes2009; 1: 111–148.
10.
RoubíčekT. Models of microstructure evolution in shape memory materials. In: Ponte CastanedaP. (eds.) Nonlinear homogenization and its application to composites, polycrystals and smart materials (NATO Science Series II, vol. 170). Dordrecht: Kluwer, 2004, 269–304.
11.
AuricchioFPetriniL. A three-dimensional model describing stress-temperature induced solid phase transformations: Solution algorithm and boundary value problems. Int J Numer Meth Eng2004; 61: 807–836.
12.
ArghavaniJAuricchioFNaghdabadiR. A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings. Int J Plast2010; 26: 976–991.
13.
BouvetCCallochSLexcellentC. A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loading. Eur J Mech A2004; 23: 37–61.
14.
ChemiskyYDuvalAPatoorE. Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation. Mech Mater2011; 43: 361–376.
15.
LagoudasDCHartlDJChemiskyY. Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. Int J Plast2012; 32–33: 155–183.
16.
PeultierBBen ZinebTPatoorE. Macroscopic constitutive law for SMA: Application to structure analysis by FEM. Mech Mater2006; 38: 510–524.
17.
ZakiWMoumniZ. A three-dimensional model of the thermomechanical behavior of shape memory alloys. J Mech Phys Solids2007; 55: 2455–2490.
18.
PanicoMBrinsonLC. A three-dimensional phenomenological model for martensite reorientation in shape memory alloys. J Mech Phys Solids2007; 55: 2491–2511.
ŠittnerPHellerLPilchJ. Roundrobin SMA modeling. In: ESOMAT 2009 – The 8th European Symposium on Martensitic Transformations, 2009.
21.
HalphenBNguyenQS. Sur les matériaux standard généralisés. J Mec1975; 14: 39–63.
22.
MielkeA. A mathematical framework for standard generalized materials in the rate-independent case. In: HelmigR. (eds.) Multifield problems in fluid and solid mechanics (Lecture Notes in Applied and Computational Mechanics, vol. 28). Berlin: Springer, 2006.
23.
FrancfortGMielkeA. Existence results for a class of rate-independent material models with nonconvex elastic energies. J Reine Angew Math2006; 595: 55–91.
MielkeATheilFLevitasVI. A variational formulation of rate-independent phase transformations using an extremum principle. Arch Rat Mech Anal2002; 162(2): 137–177.
26.
FrostMSedlákPBenešováB. Macroscopic thermomechanical model suitable for simulations of anisotropic NiTi shape memory alloys with R-phase. In: Proceedings of the International Conference on Shape Memory and Superelastic Technologies, Prague, Czech Republic, 2013, 300–301.
27.
SedlákPFrostMBenešováB. Thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings. Int J Plast2012; 39: 132–151.
28.
KochmannDMHacklK. The evolution of laminates in finite crystal plasticity: A variational approach. Continuum Mech Thermodyn2011; 23(1): 63–85.
29.
HacklKHeinzSMielkeA. A model for the evolution of laminates in finite-strain elastoplasticity. Z Angew Math Mech2012; 92(11–12): 888–909.
30.
PetrykHStupkiewiczS. Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J Mech Phys Solids2010; 58: 390–408.
31.
PetrykH. Incremental energy minimization in dissipative solids. CR Mec2003; 331: 469–474.
32.
NguyenQS. Stability and nonlinear solid mechanics. New York, NY: John Wiley, 2000.
33.
MauginGA. The thermomechanics of plasticity and fracture. Cambridge: Cambridge University Press, 1992.
34.
HacklKHeinenA. A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy. Continuum Mech Thermodyn2008; 19(8): 499–510.
35.
SouzaACMamiyaENZouainN. Three-dimensional model for solids undergoing stress-induced phase transformations. Eur J Mech A1998; 17: 789–806.
36.
SadjadpourABhattacharyaK. A micromechanics-inspired constitutive model for shape-memory alloys. Smart Mater Struct2007; 16: 1751–1765.
37.
LuigPBruhnsOT. On the modeling of shape memory alloys using tensorial internal variables. Mater Sci Eng, A2008; 481–482: 379–383.
38.
JacobusKSehitogluHBalzerM. Effect of stress state on the stress-induced martensitic transformation in polycrystalline Ni-Ti alloy. Metall Mater Trans A1996; 27: 3066–3073.
39.
OtsukaKWaymanCM. Shape memory materials. Cambridge: Cambridge University Press, 1998.
40.
OtsukaKRenX. Physical metallurgy of Ti-Ni-based shape memory alloys. Prog Mater Sci2005; 50: 511–678.
41.
OlbrichtJYawnyAPelegrinaJL. On the stress-induced formation of R-phase in ultra-fine-grained Ni-rich NiTi shape memory alloys. Metall Mater Trans A2011; 42A: 2556–2574.
42.
HeinenRMiroS. Micromechanical modeling of NiTi shape memory alloys including austenite, R-phase, and martensite. Comput Meth Appl Mech Eng2012; 229–232: 44–55.
43.
SeinerHLandaM. Non-classical austenite-martensite interfaces observed in single crystals of Cu-Al-Ni. Phase Transit2009; 82: 793–807.
44.
LiuYXieZVan HumbeeckJ. Deformation of shape memory alloys associated with twinned domain re-configurations. Mater Sci Eng, A1999; 273–275: 679–684.
45.
SeinerHSedlákPLandaM. Shape recovery mechanism observed in single crystals of Cu-Al-Ni shape memory alloy. Phase Transit2008; 81: 537–551.
46.
BernardiniDPenceTJ. Models for one-variant shape memory materials based on dissipation functions. Int J Nonlin Mech2002; 37: 1299–1317.
47.
ZhangJXSatoMIshidaA. Deformation mechanism of martensite in Ti-rich TiNi shape memory alloy thin films. Acta Mater2006; 54: 1185–1198.
48.
KockarBKaramanIKimJI. Thermomechanical cyclic response of an ultrafine-grained NiTi shape memory alloy. Acta Mater2008; 56: 3630–3646.
49.
UrbinaCDelaFlorSFerrandoF. Effect of thermal cycling on the thermomechanical behaviour of NiTi shape memory alloys. Mater Sci Eng, A2009; 501: 197–206.
50.
FrostMSedlákPSippolaM. Thermomechanical model for NiTi shape memory wires. Smart Mater Struct2010; 19: 094010.
51.
BonettiEFrémondMLexcellentC. Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain. Math Mech Solids2006; 11: 251–275.
52.
StupkiewiczSPetrykH. A robust model of pseudoelasticity in shape memory alloys. Int J Numer Meth Eng2013; 93: 747–769.
53.
ŠittnerPLandaMLukášP. R-phase transformation phenomena in thermomechanically loaded NiTi polycrystals. Mech Mater2006; 38: 475–492.
54.
FrostM. Modeling of phase transformations in shape memory materials. PhD thesis, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic, 2012.
55.
MainikAMielkeA. Existence results for energetic models for rate-independent systems. Calc Var Partial Differ Equ2005; 22(1): 73–99.
56.
MielkeA. Evolution of rate-independent systems. In: DafermosAFeireislE (eds.) Handbook of differential equations: Evolutionary differential equations. Amsterdam: Elsevier/North-Holland, 2005, 561–559.
57.
KružíkMMielkeARoubíčekT. Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi. Meccan2005; 40(5–6): 389–418.
58.
AmbrosioLFuscoNPallaraD. Functions of bounded variation and free discontinuity problems (Oxford Mathematical Monographs). Oxford: Clarendon Press, 2000.
59.
Dal MasoGDeSimoneAMoraM. A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch Rat Mech Anal2008; 189(3): 469–544.
60.
EfendievMMielkeA. On the rate-independent limit of systems with dry friction and small viscosity. J Conv Anal2006; 13(1): 151.
61.
KneesDMielkeAZaniniC. On the inviscid limit of a model for crack propagation. Math Models Meth Appl Sci2008; 18(9): 1529–1569.
62.
MielkeARossiR. Existence and uniqueness results for a class of rate-independent hysteresis problems. Math Models Meth Appl Sci2007; 17(1): 81–123.
63.
MielkeAPetrovA. Thermally driven phase transformation in shape-memory alloys. Adv Math Sci Appl2007; 17(2): 667–685.
64.
MielkeAPaoliLPetrovA. On existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys. SIAM J Math Anal2009; 41(4): 1388–1414.
65.
RoubíčekT. Thermodynamics of rate-independent processes in viscous solids at small strains. SIAM J Math Anal2010; 42(1): 256–297.
66.
BenešováBRoubíčekT. Micro-to-meso scale limit for shape-memory-alloy models with thermal coupling. Multiscale Model Simul2012; 10(3): 1059–1089.
67.
DacorognaB. Direct methods in the calculus of variations (Applied Mathematical Sciences, vol. 78). 2nd ed.Berlin: Springer-Verlag, 2008.
68.
BarbuVPrecupanuT. Convexity and optimization in Banach spaces. New York, NY: Springer, 2012.
69.
Dal MasoGFrancfortGAToaderR. Quasistatic crack growth in nonlinear elasticity. Arch Rat Mech Anal2005; 176(2): 165–225.
70.
BourdinB. Numerical implementation of the variational formulation for quasi-static brittle fracture. Interface Free Bound2007; 9(3): 411–430.
71.
BourdinBFrancfortGAMarigoJJ. The variational approach to fracture. J Elast2008; 91(1–3): 5–148.
72.
NelderJAMeadR. A simplex method for function minimization. Comput J1965; 7: 308–313.
73.
GallKSehitogluHChumlyakovYI. Tension-compression asymmetry of the stress-strain response in aged single crystal and polycrystalline NiTi. Acta Mater1999; 43: 1203–1217.
74.
GrabeCBruhnsOT. On the viscous and strain rate dependent behavior of polycrystalline NiTi. Int J Solids Struct2008; 45: 1876–1895.
75.
BartelTHacklK. A micromechanical model for martensitic phase transformations in shape memory alloys based on energy relaxation. Z Angew Math Mech2009; 89(10): 792–809.
76.
KrejčíPStefanelliU. Existence and non-existence for the full thermomechanical Souza–Auricchio model of shape memory wires. Math Mech Solids2011; 16: 349–365.
77.
HacklKFischerFD. On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc R Soc A2008; 464: 117–132.
78.
LaydiMRLexcellentC. Yield criteria for shape memory materials: Convexity conditions and surface transport. Math Mech Solids2010; 15: 165–208.
